# Non-Commutative Mechanics in Mathematical & in Condensed Matter Physics

@article{Horvathy2006NonCommutativeMI, title={Non-Commutative Mechanics in Mathematical \& in Condensed Matter Physics}, author={P. A. Horvathy}, journal={Symmetry Integrability and Geometry-methods and Applications}, year={2006}, volume={2}, pages={090} }

Non-commutative structures were introduced, independently and around the same time, in mathematical and in condensed matter physics (see Table 1). Souriau’s construction applied to the two-parameter central extension of the planar Galilei group leads to the “exotic” particle, which has non-commuting position coordinates. A Berryphase argument applied to the Bloch electron yields in turn a semiclassical model that has been used to explain the anomalous/spin/optical Hall effects. The non…

## 13 Citations

### Super)symmetries of semiclassical models in theoretical and condensed matter physics

- Physics
- 2011

Van Holten’s covariant algorithm for deriving conserved quantities is presented, with particular attention paid to Runge-Lenz-type vectors. The classical dynamics of isospin-carrying particles is…

### Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism

- Physics, Mathematics
- 2015

We consider Feynman-Dyson’s proof of Maxwell’s equations u sing the Jacobi identities on the velocity phase space. In this paper we generalize the Feynman-Dyson’s scheme by incorporating the…

### Exotic Newton-Hooke Group, Noncommutative Plane and Superconformal Symmetry

- Physics, Mathematics
- 2009

In this thesis some systems with exotic symmetries, which are a peculiarity in 2+1 space-time dimensions, are studied. Coded in the exotic structure, there appear noncommutative coordinates and a…

### Comments on Galilean conformal field theories and their geometric realization

- MathematicsJournal of High Energy Physics
- 2010

We discuss non-relativistic conformal algebras generalizing the Schrödinger algebra. One instance of these algebras is a conformal, acceleration-extended, Galilei algebra, which arises also as a…

### Exotic Non-relativistic String

- Physics
- 2013

We construct a classical non-relativistic string model in 3+1 dimensions. The model contains a spurion tensor ﬁeld that is responsible for the non-commutative structure of the model. Under double…

### Operator Deformations in Quantum Measurement Theory

- Physics
- 2014

In this paper, we develop a rigorous observable- and symmetry generator-related framework for quantum measurement theory by applying operator deformation techniques previously used in noncommutative…

### Operator Deformations in Quantum Measurement Theory

- PhysicsLetters in Mathematical Physics
- 2013

In this paper, we develop a rigorous observable- and symmetry generator-related framework for quantum measurement theory by applying operator deformation techniques previously used in noncommutative…

### A note on the Chevalley–Eilenberg cohomology for the Galilei and Poincaré algebras

- Mathematics
- 2009

We construct in a systematic way the complete Chevalley–Eilenberg cohomology at form degrees 2, 3 and 4 for the Galilei and Poincaré groups. The corresponding non-trivial forms belong to certain…

### Exotic nonrelativistic string

- Physics
- 2007

We construct a classical nonrelativistic string model in 3+1 dimensions. The model contains a spurion tensor field that is responsible for the noncommutative structure of the model. Under…

## References

SHOWING 1-10 OF 50 REFERENCES

### monopole and Berry phase in momentum space in noncommutative quantum mechanics

- Physics
- 2004

To build genuine generators of the rotations group in noncommutative quantum mechanics, we show that it is necessary to extend the noncommutative parameter $\ensuremath{\theta}$ to a field operator,…

### Exotic Galilean symmetry in the non-commutative plane and the Hall effect

- Physics
- 2001

Quantum mechanics in the non-commutative plane is shown to admit the 'exotic' symmetry of the doubly centrally extended Galilei group. When coupled to a planar magnetic field whose strength is the…

### The Non-commutative Landau Problem

- Physics
- 2002

Abstract The Landau problem is discussed in two similar but still different non-commutative frameworks. The “standard” one, where the coupling to the gauge field is achieved using Poisson brackets,…

### "Topological" (Chern-Simons) quantum mechanics.

- PhysicsPhysical review. D, Particles and fields
- 1990

Two quantum-mechanical models are constructed that are analogs of three-dimensional, topologically massive as well as Chern-Simons gauge-field theories, and the phase-space reductive limiting procedure that takes the former to the latter is studied.

### 2 0 Se p 20 06 Anomalous Hall Effect in non-commutative mechanics

- Physics
- 2008

The anomalous velocity term in the semiclassical model of a Bloch electron deviates the trajectory from the conventional one. When the Berry curvature (alias noncommutative parameter) is a monopole…

### Noncommuting coordinates in the Hall effect and in vortex dynamics

- Physics
- 2003

Laughlin's Ansatz to explain the fractional Quantum Hall effect is derived by coupling a particle associated with ``exotic'' the two-fold central extension of the planar Galilei group. The reduced…

### Noncommuting Coordinates, Exotic Particles, and Anomalous Anyons in the Hall Effect

- Physics
- 2004

We review our previous “exotic” particle, together with the more recent anomalous anyon model (which has the arbitrary gyromagnetic factor g). The nonrelativistic limit of the anyon generalizes the…